Dual-bounded generating problems: weighted transversals of a hypergraph

We consider a generalization of the notion of transversal to a finite hypergraph, so called weighted transversals. Given a non-negative weight vector assigned to each hyperedge of an input hypergraph A and a nonnegative threshold vector, we define a weighted transversal as a minimal vertex set which intersects all the hyperedges of A except for a subfamily of total weight not exceeding the given threshold vector. Weighted transversals generalize partial and multiple transversals introduced in [8] and also include minimal binary solutions to non-negative systems of linear inequalities and minimal weighted infrequent sets in databases. We show that the hypergraph of all weighted transversals is dual-bounded, i.e, the size of its dual hypergraph is polynomial in the number of weighted transversals and the size of the input hypergraph. Our bounds are based on new inequalities of extremal set theory and threshold Boolean logic, which may be of independent interest. For instance, we show that that for any row-weighted m × n binary matrix and any threshold weight t, the number of maximal sets of columns whose row support has weight above t is at most m times the number of minimal sets of columns with row support of total weight below t. We also prove that the problem of generating all weighted transversals for a given hypergraph is polynomialtime reducible to the generation of all ordinary transversals for another The research of the first two authors was supported in part by the Office of Naval Research (Grant N00014-92-J-1375), and the National Science Foundation (Grant DMS 98-06389). Visits of the second author to Rutgers University were also supported by DIMACS, the National Science Foundation’s Center for Discrete Mathematics and Theoretical Computer Science. The research of the third author was supported in part by the National Science Foundation (Grant CCR-9618796). The research of the fourth author was supported in part by a Scientific Grant-in-Aid by the Ministry of Education, Science, Sports, and Culture of Japan. A preliminary version of this paper appeared in [9] RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway NJ 08854-8003; ([email protected]). RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway NJ 08854-8003; ([email protected]) Department of Computer Science, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854; ([email protected]) Division of Systems Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, 560-8531, Japan; ([email protected])